New York Journal of Mathematics
Volume 18 (2012) 621-650


Antun Milas and Michael Penn

Lattice vertex algebras and combinatorial bases: general case and W-algebras

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Published: August 24, 2012
Keywords: Vertex operator algebras, integral lattices
Subject: 17B69,17B67, 11P81

We introduce what we call the principal subalgebra of a lattice vertex (super) algebra associated to an arbitrary Z-basis of the lattice. In the first part (to appear), the second author considered the case of positive bases and found a description of the principal subalgebra in terms of generators and relations. Here, in the most general case, we obtain a combinatorial basis of the principal subalgebra WL and of related modules. In particular, we substantially generalize several results in Georgiev, 1996, covering the case of the root lattice of type An, as well as some results from Calinescu, Lepowsky and Milas, 2010. We also discuss principal subalgebras inside certain extensions of affine W-algebras coming from multiples of the root lattice of type An.


The first author graciously acknowledges support from NSA and NSF grants.

Author information

Antun Milas:
Max Planck Institute für Mathematik, Vivatsgasse 7, Bonn, Germany
Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY 12222

Michael Penn:
Department of Mathematics, University of Tennessee, Chattanooga